Calabi--Yau Compactification and Hodge Data

A ten-dimensional string background becomes a four-dimensional theory after choosing a six-dimensional compact internal space. In the simplest flux-free ansatz,

$$X_{10}=M^{3,1}\times X,$$

where $X$ is compact and small compared with the length scales probed by a four-dimensional observer. The central question is: what kind of six-manifold can preserve supersymmetry while solving the leading low-energy equations of motion?

For type II strings the standard answer is a Calabi—Yau threefold. The word threefold means complex dimension three, hence real dimension six. A compactification on such an $X$ gives a four-dimensional theory with eight real supercharges,

$$\text{type II on a Calabi--Yau threefold} \quad\Longrightarrow\quad 4d\,\mathcal N=2.$$

The wonderful economy of the subject is that much of the massless spectrum is governed by two integers,

$$h^{1,1}(X), \qquad h^{2,1}(X).$$

The first counts Kähler deformations; the second counts complex-structure deformations. Type IIA and type IIB place these moduli in different four-dimensional multiplets, and mirror symmetry exchanges the two numbers.

The geometry in one paragraph

A Calabi—Yau threefold is a compact Kähler complex threefold with vanishing first Chern class. Equivalently, for the generic smooth case relevant to type II compactification, its holonomy is $SU(3)$. The condition $c_1(X)=0$ implies that the canonical bundle is trivial, so there is a nowhere-vanishing holomorphic three-form

$$\Omega\in H^{3,0}(X).$$

By Yau’s theorem, every Kähler class contains a unique Ricci-flat Kähler metric. The $SU(3)$ holonomy is the spacetime supersymmetry condition: it leaves one internal spinor invariant.

The two basic differential forms are

$$J=i g_{i\bar j}dz^i\wedge d\bar z^{\bar j}, \qquad \Omega=\Omega_{ijk}dz^i\wedge dz^j\wedge dz^k,$$

where $J$ is the Kähler form and $\Omega$ is the holomorphic volume form. They obey compatibility conditions of the schematic form

$$J\wedge\Omega=0, \qquad {i\over 8}\Omega\wedge\bar\Omega={1\over 3!}J^3,$$

after a conventional normalization. The form $J$ measures Kähler data such as volumes of holomorphic cycles; the form $\Omega$ measures complex-structure data through its periods over three-cycles.

Kähler geometry from the worldsheet viewpoint

The internal part of the string sigma model contains

$$S_X={1\over 4\pi\alpha'}\int d^2\sigma\sqrt h\,h^{ab} G_{mn}(Y)\partial_aY^m\partial_bY^n+\cdots.$$

Conformal invariance of this two-dimensional theory is the string equation of motion. At leading order in $\alpha'$, the metric beta function is

$$\beta^G_{mn}=\alpha' R_{mn}+O(\alpha'^2).$$

Thus a purely metric compactification wants a Ricci-flat internal metric. Supersymmetry requires more: the target must admit a complex structure compatible with the metric. Let $I$ be the complex structure. It satisfies

$$I^2=-1, \qquad g(IX,IY)=g(X,Y).$$

The associated two-form is

$$J(X,Y)=g(IX,Y).$$

The Kähler condition is

$$dJ=0.$$

In local holomorphic coordinates, a Kähler metric is locally determined by a Kähler potential $K$:

$$g_{i\bar j}=\partial_i\partial_{\bar j}K.$$

This is the geometry naturally selected by the worldsheet theory with extended supersymmetry.

Holonomy and preserved supersymmetry

The ten-dimensional supersymmetry parameter decomposes as a four-dimensional spinor times an internal spinor,

$$\epsilon_{10}=\epsilon_4\otimes\eta_6+\text{complex conjugate}.$$

Unbroken supersymmetry requires

$$\nabla_m\eta_6=0.$$

For a generic six-dimensional Riemannian manifold the holonomy is $SO(6)\simeq SU(4)$. A chiral spinor transforms in the $\mathbf 4$ of $SU(4)$. Under $SU(3)\subset SU(4)$,

$$\mathbf 4\longrightarrow \mathbf 3\oplus \mathbf 1.$$

The singlet is the covariantly constant internal spinor. Type II string theory has two ten-dimensional supersymmetry parameters, so compactification on a Calabi—Yau threefold produces two four-dimensional minimal supersymmetries: $\mathcal N=2$ in four dimensions. A smaller holonomy group, such as the holonomy of a torus or $K3\times T^2$, would preserve more supersymmetry.

The Hodge diamond of a smooth compact Calabi—Yau threefold with holonomy $SU(3)$. All entries are fixed by the two independent Hodge numbers $h^{1,1}$ and $h^{2,1}$.

Hodge decomposition and the Hodge diamond

On a compact Kähler manifold, complex differential forms decompose into types $(p,q)$, and the cohomology decomposes as

$$H^k(X,\mathbb C)=\bigoplus_{p+q=k}H^{p,q}(X).$$

The Hodge numbers are

$$h^{p,q}=\dim H^{p,q}(X).$$

For a compact Kähler threefold,

$$h^{p,q}=h^{q,p}, \qquad h^{p,q}=h^{3-p,3-q}.$$

The first relation is complex conjugation; the second is Poincaré duality combined with the Hodge star. For a Calabi—Yau threefold with full $SU(3)$ holonomy,

$$h^{0,0}=h^{3,3}=1, \qquad h^{3,0}=h^{0,3}=1,$$

and

$$h^{1,0}=h^{0,1}=h^{2,0}=h^{0,2}=0.$$

Therefore the Hodge diamond is

$$\begin{matrix} &&&1&&&\\ &&0&&0&&\\ &0&&h^{1,1}&&0&\\ 1&&h^{2,1}&&h^{2,1}&&1\\ &0&&h^{1,1}&&0&\\ &&0&&0&&\\ &&&1&&& \end{matrix}$$

The Betti numbers are obtained by summing along the diagonals,

$$b_k=\sum_{p+q=k}h^{p,q}.$$

Thus

$$\begin{aligned} b_0&=b_6=1,\\ b_1&=b_5=0,\\ b_2&=b_4=h^{1,1},\\ b_3&=2h^{2,1}+2. \end{aligned}$$

The Euler characteristic is

$$\chi(X)=\sum_{k=0}^6(-1)^k b_k =2\bigl(h^{1,1}-h^{2,1}\bigr).$$

This formula is a useful first check on proposed mirror pairs, because mirror symmetry exchanges $h^{1,1}$ and $h^{2,1}$ and therefore reverses the sign of $\chi$.

Kähler moduli

Choose a basis $\{\omega_a\}$ of harmonic $(1,1)$ forms,

$$a=1,\ldots,h^{1,1}.$$

The Kähler form expands as

$$J=t^a\omega_a.$$

The real parameters $t^a$ measure volumes of two-cycles in a dual basis. The NS—NS two-form has zero modes along the same harmonic forms,

$$B_2=b^a\omega_a.$$

Together they form complexified Kähler moduli,

$$T^a=b^a+i t^a.$$

The total volume is

$$\mathcal V(X)={1\over 3!}\int_XJ\wedge J\wedge J ={1\over6}\kappa_{abc}t^at^bt^c,$$

where the triple intersection numbers are

$$\kappa_{abc}=\int_X\omega_a\wedge\omega_b\wedge\omega_c.$$

At large radius these intersection numbers determine the leading type IIA vector-multiplet prepotential,

$$\mathcal F_{\rm IIA}(T)=-{1\over6}\kappa_{abc}T^aT^bT^c+\cdots.$$

The ellipsis includes lower-degree terms and worldsheet instanton corrections.

Complex-structure moduli

Complex-structure deformations change which local coordinates are holomorphic. Infinitesimally they are described by Beltrami differentials,

$$\mu=\mu^i{}_{\bar j}{\partial\over \partial z^i}\otimes d\bar z^{\bar j}.$$

The deformed holomorphic one-forms are schematically

$$dz^i\longrightarrow dz^i+\mu^i{}_{\bar j}d\bar z^{\bar j}.$$

To first order, integrability gives $\bar\partial\mu=0$, and changes by $\bar\partial$ of a vector field are gauge redundancies. Thus infinitesimal complex-structure deformations are counted by

$$H^1(X,T_X).$$

The holomorphic three-form converts this into Dolbeault cohomology of forms. Contracting the vector index of $\mu$ into $\Omega$ gives

$$H^1(X,T_X)\simeq H^{2,1}(X).$$

Therefore the number of complex-structure moduli is $h^{2,1}$. Equivalently, a variation of the holomorphic three-form has the form

$$\partial_I\Omega=k_I\Omega+\chi_I, \qquad \chi_I\in H^{2,1}(X), \qquad I=1,\ldots,h^{2,1}.$$

The periods of $\Omega$ over a symplectic basis of three-cycles are especially important:

$$X^I=\int_{A^I}\Omega, \qquad \mathcal F_I=\int_{B_I}\Omega, \qquad I=0,\ldots,h^{2,1}.$$

Only projective ratios of periods are physical complex-structure coordinates, since rescaling $\Omega$ does not change the complex structure. The corresponding special Kähler potential is

$$K_{\rm cs}=-\log\left(i\int_X\Omega\wedge\bar\Omega\right),$$

up to convention-dependent constants.

Massless fields from harmonic forms

A ten-dimensional field can be expanded in internal eigenmodes,

$$\Phi(x,y)=\sum_n\phi_n(x)Y_n(y).$$

The four-dimensional mass of $\phi_n$ is determined by the internal Laplacian eigenvalue of $Y_n$. Zero modes are harmonic forms, so cohomology counts massless fields.

This is why the Hodge numbers enter the four-dimensional spectrum. Metric zero modes preserving Ricci-flatness split into Kähler and complex-structure deformations. Form fields reduce on harmonic forms: harmonic two-forms give zero modes of $B_2$ and RR potentials, while harmonic three-forms encode period data and RR scalars. In pure type II Calabi—Yau compactification these moduli are massless. Fluxes, orientifolds, branes, and nonperturbative effects can lift them, but those ingredients are beyond this simplest setting.

Type IIA and type IIB compactified on the same Calabi—Yau threefold distribute Kähler and complex-structure moduli differently among four-dimensional $\mathcal N=2$ multiplets. Mirror symmetry exchanges $h^{1,1}$ and $h^{2,1}$.

Type IIA versus type IIB

A four-dimensional $\mathcal N=2$ vector multiplet contains one vector and one complex scalar. A hypermultiplet contains four real scalars. The gravity multiplet contains the metric, gravitini, and a graviphoton.

For type IIA on $X$, the vector multiplet scalars are the complexified Kähler moduli:

$$n_V^{\rm IIA}=h^{1,1}(X).$$

The hypermultiplets contain the complex-structure moduli and one universal hypermultiplet:

$$n_H^{\rm IIA}=h^{2,1}(X)+1.$$

For type IIB on $X$, the assignments are exchanged:

$$n_V^{\rm IIB}=h^{2,1}(X), \qquad n_H^{\rm IIB}=h^{1,1}(X)+1.$$

The $+1$ is the universal hypermultiplet, containing the four-dimensional dilaton together with axionic partners. In real scalar counts,

$$\begin{array}{c|c|c} \text{theory on }X & \text{vector-multiplet scalars} & \text{hypermultiplet scalars}\\ \hline \text{IIA} & 2h^{1,1} & 4(h^{2,1}+1)\\ \text{IIB} & 2h^{2,1} & 4(h^{1,1}+1) \end{array}$$

This split is the easiest way to remember the type II Calabi—Yau spectrum.

Mirror symmetry

A mirror pair of Calabi—Yau threefolds $(X,X^\vee)$ satisfies

$$h^{1,1}(X)=h^{2,1}(X^\vee), \qquad h^{2,1}(X)=h^{1,1}(X^\vee).$$

Consequently,

$$\chi(X^\vee)=-\chi(X).$$

The physical statement is stronger than equality of Hodge data:

$$\boxed{\text{type IIA on }X\quad\simeq\quad\text{type IIB on }X^\vee.}$$

Under this equivalence,

$$\text{Kähler moduli of }X \quad\longleftrightarrow\quad \text{complex-structure moduli of }X^\vee.$$

This is powerful because quantum corrections to Kähler moduli on one side, such as worldsheet instantons, can be translated into classical period calculations for the complex structure of the mirror.

The conifold preview

Smooth Calabi—Yau moduli spaces can have boundaries where cycles shrink. At such points extra branes can become light, and the naive low-energy description becomes singular. The simplest local model is the conifold,

$$\sum_{a=1}^4 z_a^2=0 \qquad \text{inside }\mathbb C^4,$$

or equivalently

$$xy-uv=0.$$

It is one complex equation inside $\mathbb C^4$, so it is a complex threefold. Away from the origin it is a cone,

$$ds_6^2=dr^2+r^2ds^2_{T^{1,1}},$$

with base

$$T^{1,1}={SU(2)\times SU(2)\over U(1)}.$$

The conifold is the local Calabi—Yau singularity behind the next AdS/CFT example. Placing D3-branes at the tip gives a near-horizon geometry $AdS_5\times T^{1,1}$ and a four-dimensional $\mathcal N=1$ superconformal quiver gauge theory.

The conifold is a noncompact Calabi—Yau cone. Its base is $T^{1,1}$, the internal space of the Klebanov—Witten $AdS_5\times T^{1,1}$ duality.

Summary

For a Calabi—Yau threefold,

$$\begin{array}{c|c} \text{geometric datum} & \text{count}\\ \hline \text{Kähler moduli} & h^{1,1}\\ \text{complex-structure moduli} & h^{2,1}\\ \text{three-cycles} & b_3=2h^{2,1}+2\\ \text{Euler characteristic} & \chi=2(h^{1,1}-h^{2,1}) \end{array}$$

Type IIA puts the Kähler moduli in vector multiplets and the complex-structure moduli in hypermultiplets. Type IIB does the opposite. Mirror symmetry exchanges $h^{1,1}$ and $h^{2,1}$ and turns this exchange into a physical equivalence.

Exercises

Exercise 1. Holonomy and supersymmetry

Use

$$\mathbf 4\to \mathbf 3\oplus \mathbf 1$$

under $SU(4)\to SU(3)$ to explain why type II compactification on a Calabi—Yau threefold preserves eight real supercharges in four dimensions.

Solution

A covariantly constant spinor must be invariant under the holonomy group. In six Euclidean dimensions, $Spin(6)\simeq SU(4)$, and a chiral spinor transforms as $\mathbf 4$. Under $SU(3)\subset SU(4)$,

$$\mathbf 4\to \mathbf 3\oplus \mathbf 1.$$

The singlet gives one invariant internal spinor. Type II string theory has two ten-dimensional supersymmetry parameters, and each can use this internal singlet to produce four-dimensional supersymmetry. The result is four-dimensional $\mathcal N=2$, or eight real supercharges.

Exercise 2. Betti numbers from the Hodge diamond

Show that a Calabi—Yau threefold has

$$b_3=2h^{2,1}+2, \qquad \chi=2(h^{1,1}-h^{2,1}).$$

Solution

The third Betti number is

$$b_3=h^{3,0}+h^{2,1}+h^{1,2}+h^{0,3}.$$

Using $h^{3,0}=h^{0,3}=1$ and $h^{1,2}=h^{2,1}$, we get

$$b_3=2h^{2,1}+2.$$

The other Betti numbers are

$$b_0=b_6=1, \quad b_1=b_5=0, \quad b_2=b_4=h^{1,1}.$$

Therefore

$$\chi=1+h^{1,1}-(2h^{2,1}+2)+h^{1,1}+1 =2(h^{1,1}-h^{2,1}).$$

Exercise 3. The quintic test case

The quintic threefold has

$$h^{1,1}=1, \qquad h^{2,1}=101.$$

Compute $b_2$, $b_3$, and $\chi$.

Solution

The second Betti number is

$$b_2=h^{1,1}=1.$$

The third Betti number is

$$b_3=2h^{2,1}+2=2\cdot 101+2=204.$$

The Euler characteristic is

$$\chi=2(h^{1,1}-h^{2,1})=2(1-101)=-200.$$

Exercise 4. Volume and triple intersections

Let $J=t^a\omega_a$ and

$$\kappa_{abc}=\int_X\omega_a\wedge\omega_b\wedge\omega_c.$$

Derive

$$\mathcal V={1\over6}\kappa_{abc}t^at^bt^c.$$

Solution

For a Kähler threefold, the volume form is

$${1\over3!}J\wedge J\wedge J.$$

Thus

$$\mathcal V={1\over3!}\int_XJ^3.$$

Substituting $J=t^a\omega_a$ gives

$$\mathcal V={1\over6}t^at^bt^c\int_X\omega_a\wedge\omega_b\wedge\omega_c ={1\over6}\kappa_{abc}t^at^bt^c.$$

Exercise 5. Complex-structure moduli

Explain why infinitesimal complex-structure deformations on a Calabi—Yau threefold are counted by $h^{2,1}$.

Solution

An infinitesimal complex-structure deformation is an element of $H^1(X,T_X)$. The holomorphic three-form converts the vector index of such a deformation into two holomorphic form indices:

$$\mu^i{}_{\bar j}d\bar z^{\bar j}\otimes \partial_i \quad\mapsto\quad \mu^i{}_{\bar j}\Omega_{ikl}d\bar z^{\bar j}\wedge dz^k\wedge dz^l.$$

The result is a $(2,1)$ form. Thus

$$H^1(X,T_X)\simeq H^{2,1}(X),$$

so the number of complex-structure moduli is $h^{2,1}$.

Exercise 6. Type II multiplet counts

For a Calabi—Yau threefold with $(h^{1,1},h^{2,1})=(3,75)$, find the numbers of vector multiplets and hypermultiplets for type IIA and type IIB compactifications.

Solution

For type IIA,

$$n_V=h^{1,1}=3, \qquad n_H=h^{2,1}+1=76.$$

For type IIB,

$$n_V=h^{2,1}=75, \qquad n_H=h^{1,1}+1=4.$$

The $+1$ is the universal hypermultiplet.

Exercise 7. Dimension of the conifold base

Consider the conifold $xy-uv=0$ inside $\mathbb C^4$. Show that it is a real six-dimensional cone and that its base is real five-dimensional.

Solution

The ambient space $\mathbb C^4$ has real dimension $8$. The conifold equation is one complex equation, so it imposes two real constraints. Therefore the conifold has real dimension

$$8-2=6.$$

The equation is homogeneous: if $(x,y,u,v)$ is a solution, then so is $(\lambda x,\lambda y,\lambda u,\lambda v)$. This gives a radial cone direction. Fixing

$$r^2=|x|^2+|y|^2+|u|^2+|v|^2$$

removes one real dimension, leaving a real five-dimensional base. For the conifold, this base is $T^{1,1}$.